Problem 89 - College Algebra Essentials

Question

Begin by graphing the square root function, \(f(x)=\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$g(x)=-|x+4|+1$$

Step-by-Step Solution

Answer

The graph of the square root function, \(f(x) = \sqrt{x}\) starts from the point (0,0) and increases. To graph \(g(x) = -|x+4|+1\), begin from the point (-4,1) and have it decrease, mirroring the increase of the square root function below the x-axis due to the reflection. This graph would then be translated 1 unit upwards.

Step 1: Graph the Square Root Function

Begin by graphing the square root function \(f(x) = \sqrt{x}\). You can start plotting values of x starting from 0 and calculate the corresponding y values as the square root of x, plotting these points on a graph.

Step 2: Identify the Transformations

The function \(g(x) = -|x+4|+1\) undergoes two transformations. The negative sign implies a reflection over the x-axis while +4 and +1 refer to a horizontal and vertical shift respectively. Subtracting 4 from x shifts the graph 4 units to the left whereas adding 1 to y, shifts it one unit upwards.

Step 3: Apply the Transformations and Graph

Now apply these transformations to the graph of \(f(x) = \sqrt{x}\) to graph \(g(x) = -|x+4|+1\). Begin from the point (0,0) and shift 4 units to the left to (-4,0). Since there's a reflection over the x-axis, vertical values will be inverted. Thus, given the absolute value, all y-values will be non-positive. Then, shift every point on the graph 1 unit upwards due to the +1 in the function.

Key Concepts

Square Root Function

The square root function is represented mathematically as \( f(x) = \sqrt{x} \). This is a basic function where outputs are the square roots of their inputs. Its graph is a curve that starts at the origin (0,0) and gradually increases as it moves to the right. This reflects that as x becomes larger, the value of \( \sqrt{x} \) also increases, but at a decreasing rate. The graph of a square root function is always non-negative, showcasing a range limited from 0 upwards.

• Starts from the origin: (0,0)
• Values only for non-negative x due to the nature of square roots
• Gradual increase along the x-axis

By understanding how to plot this, students have a foundation for applying transformations.

Absolute Value Function

The absolute value function is unique because it reflects both positive and negative inputs into positive outputs. Written as \( |x| \), it represents the distance from zero on the number line, always non-negative. Its graph is a V-shaped plot centered at the origin. Each side of the V is a line: the left side is the reflection of the right about the y-axis.

• Key point: the origin (0,0)
• Graph symmetry about the y-axis
• Shape of a 'V'

Understanding the absolute value function helps visualize how reflections and shifts change its position and orientation.

Reflection Across the X-axis

Reflecting a graph across the x-axis means flipping it over the x-axis. This transformation changes the sign of all y-values. In mathematical terms, this involves multiplying the function by -1. When applied to any function \( h(x) \), the reflection is represented as \( -h(x) \).

• Every point \((x, y)\) becomes \((x, -y)\)
• Key visual: graph flips upside down
• No change in shape, only orientation

For the function \( g(x) = -|x+4|+1 \), this reflection is a crucial step, turning the V-shape of the absolute value upside down.

Horizontal and Vertical Shifts

Transformations involving shifts change the position of the graph on the coordinate plane. A horizontal shift involves moving the graph left or right. This is determined by changes within the function parentheses: for \( g(x) = -|x+4|+1 \), the \(+4\) represents a shift 4 units to the left.Vertical shifts move graphs up or down, depending on constants added outside function parentheses. In our example, \(+1\) shifts the graph one unit upwards.

• Horizontal shifts affect the x-value: left for \(+\), right for \(-\)
• Vertical shifts affect the y-value: up for \(+\), down for \(-\)
• Helps in positioning the graph accurately

Mastering shifts helps in understanding complex transformations like that of mixing reflections with shifts.